Why is Blender not letting me move an object that has been keyframed?
This is a very odd issue which seems to be caused by the 'Import Images as Planes' add-on. Disable the add-on in the Preferences (and re-enable it if you still want to use it) and the issue should be gone.

That add-on has some code that runs when an object moves, and somehow it's got into a broken state and is trying to update the object positions when you're trying to move them. The problem seems to keep occurring in different versions of blender because the properties that are related to the broken state are stored in your file. Disabling the add-on removes these properties and if it is re-enabled it will set them up afresh in a working state. At least, that's what seems to be happening.



By the way, while that does solve that issue, there are still some other issues in your file, related to the drivers you have set up in the compositor in each of your scenes (which may have been set up by the add-on?). The System Console (Window> Toggle System Console for Windows users) prints out a lot of errors when you change scenes and a lot of the drivers seem to have missing references to objects.